15 Cooperative Effects in Plasmonics
527
The principal difference between plasmonic and usual (photonic) dipole coupling
stems from the presence of non-radiative energy transfer between the dipoles and
the nanostructure in the former case. Let us first outline its role for the case of
a single dipole near metal NP. When an excited emitter is located close to metal
surface, its energy can be transferred to optically inactive excitations in the metal
and eventually dissipated (Ohmic losses). This is described by the non-radiative
decay rate, Γ nr ∼ d −3 , where d is the dipole–surface separation [14]. Note that very
close to metal surface (∪1 nm), this dependence changes to ∼d −4 due to surfaceassisted generation of electron–hole pairs out of the Fermi sea [15, 16], although the
non-local effects can change this behavior significantly [17]. As a result, the radiation
of a coupled dipole–NP system is governed by a competition between non-radiative
losses and plasmon enhancement [18] that determines system quantum efficiency,
Q = Γ r /Γ , where Γ = Γ r + Γ nr is the full decay rate. Indeed, the radiated energy
is W = (kc/2)Q, k and c being wave vector and speed of light, and its distance
dependence follows that of Q. Namely, with decreasing d, the emission first increases
due to the plasmon enhancement, and then, at several nm from metal surface, it is
quenched due to the suppression of Q by non-radiative losses. Both enhancement and
quenching were observed in recent experiments on fluorescing molecules attached
to a metal NP [19–23], and, not too close to NP surface, the distance dependence of
single-molecule fluorescence [21, 22] was found in excellent agreement with single
dipole–NP models [24–27].
When radiation takes place from an ensemble of emitters near a metal nanostructure, there are two distinct types of plasmon-induced couplings between the
emitters. The first is plasmon-enhanced radiative coupling, described by radiative
decay matrix Γ r
jk , where indexes j, k = 1, . . . , N refer to emitters, that is a straightforward extension of Dicke radiative coupling obtained by incorporating SP local
field. Correspondingly, the eigenstates of Γ r
jk are superradiant and subradiant states
characterized by the strength of their coupling to radiation field. In the ideal case of
“point sample,” i.e., k L ∝ 1, the subradiant decay rates are negligibly small and Γ r
jk
essentially has just three non-zero eigenvalues, corresponding to superradiant decay
rates, each scaling with N as ∪N Γ r /3 [12, 13].
The second coupling mechanism is non-radiative energy transfer between dipoles
that takes place in two steps: an excited dipole first transfers its energy to the plasmons in a nanostructure via its near field, and then this energy is transferred to
another dipole. This process involves plasmons with all angular momenta l, and it
is described by non-radiative decay matrix, Γ nr
jk . Importantly, due to plasmons with
angular momenta l > 1, the eigenstates of full decay matrix, Γ jk = Γ r
jk + Γ nr
jk , are
not superradiant and subradiant states, but their admixtures. Close to metal surface
where non-radiative processes are dominant, Γ nr
jk prevails over Γ r
jk , and no cooperative behavior is expected. However, when dipoles are separated from the surface by
more than several nm, the coupling takes place primarily via dipole (l = 1) SP and
so superradiance remains intact.
This observation is confirmed by numerical calculation of eigenvalues of Γ jk , i.e.,
full decay rates of system eigenstates, for ensemble of N dipoles near a spherical
527
The principal difference between plasmonic and usual (photonic) dipole coupling
stems from the presence of non-radiative energy transfer between the dipoles and
the nanostructure in the former case. Let us first outline its role for the case of
a single dipole near metal NP. When an excited emitter is located close to metal
surface, its energy can be transferred to optically inactive excitations in the metal
and eventually dissipated (Ohmic losses). This is described by the non-radiative
decay rate, Γ nr ∼ d −3 , where d is the dipole–surface separation [14]. Note that very
close to metal surface (∪1 nm), this dependence changes to ∼d −4 due to surfaceassisted generation of electron–hole pairs out of the Fermi sea [15, 16], although the
non-local effects can change this behavior significantly [17]. As a result, the radiation
of a coupled dipole–NP system is governed by a competition between non-radiative
losses and plasmon enhancement [18] that determines system quantum efficiency,
Q = Γ r /Γ , where Γ = Γ r + Γ nr is the full decay rate. Indeed, the radiated energy
is W = (kc/2)Q, k and c being wave vector and speed of light, and its distance
dependence follows that of Q. Namely, with decreasing d, the emission first increases
due to the plasmon enhancement, and then, at several nm from metal surface, it is
quenched due to the suppression of Q by non-radiative losses. Both enhancement and
quenching were observed in recent experiments on fluorescing molecules attached
to a metal NP [19–23], and, not too close to NP surface, the distance dependence of
single-molecule fluorescence [21, 22] was found in excellent agreement with single
dipole–NP models [24–27].
When radiation takes place from an ensemble of emitters near a metal nanostructure, there are two distinct types of plasmon-induced couplings between the
emitters. The first is plasmon-enhanced radiative coupling, described by radiative
decay matrix Γ r
jk , where indexes j, k = 1, . . . , N refer to emitters, that is a straightforward extension of Dicke radiative coupling obtained by incorporating SP local
field. Correspondingly, the eigenstates of Γ r
jk are superradiant and subradiant states
characterized by the strength of their coupling to radiation field. In the ideal case of
“point sample,” i.e., k L ∝ 1, the subradiant decay rates are negligibly small and Γ r
jk
essentially has just three non-zero eigenvalues, corresponding to superradiant decay
rates, each scaling with N as ∪N Γ r /3 [12, 13].
The second coupling mechanism is non-radiative energy transfer between dipoles
that takes place in two steps: an excited dipole first transfers its energy to the plasmons in a nanostructure via its near field, and then this energy is transferred to
another dipole. This process involves plasmons with all angular momenta l, and it
is described by non-radiative decay matrix, Γ nr
jk . Importantly, due to plasmons with
angular momenta l > 1, the eigenstates of full decay matrix, Γ jk = Γ r
jk + Γ nr
jk , are
not superradiant and subradiant states, but their admixtures. Close to metal surface
where non-radiative processes are dominant, Γ nr
jk prevails over Γ r
jk , and no cooperative behavior is expected. However, when dipoles are separated from the surface by
more than several nm, the coupling takes place primarily via dipole (l = 1) SP and
so superradiance remains intact.
This observation is confirmed by numerical calculation of eigenvalues of Γ jk , i.e.,
full decay rates of system eigenstates, for ensemble of N dipoles near a spherical